Mean curvature flow with generic initial data (Q6560723)

Mean curvature flow is a geometric heat equation, and a family of surfaces \(M(t)\) evolves by mean curvature flow if\N\[\N\left(\partial_t\mathbf{x}\right)^{\perp}=\mathbf{H}_{M(t)}(\mathbf{x}),\N\]\Nwhere \(\mathbf{H}_{M(t)}(\mathbf{x})\) denotes the mean curvature of \(M(t)\). As singularities occur in general along the mean curvature flow, a fundamental problem is to understand the singularities of mean curvature flow.\N\NThe authors obtain two main theorems which show that a large class of unstable singularity models are avoidable by a slight perturbation of the initial data. Firstly, the authors prove that if the initial closed connected hypersurface \(M^4\subset \mathbb{R}^4\) has entropy \(\lambda(M)\leq \lambda(\mathbb{S}^2\times \mathbb{R})\), then there exist arbitrary small \(C^\infty\) graphs \(M'\) over \(M\) such that the mean curvature flow starting from \(M'\) is smooth until it becomes extinct into a round point. An immediate corollary of this result is a direct proof of the low-entropy Schoenflies conjecture recently proved by \textit{J. Bernstein} and \textit{L. Wang} [Duke Math. J. 171, No. 7, 1531--1558 (2022; Zbl 1500.53093)].\N\NSecondly, the authors prove more for generic mean curvature flow of embedded surfaces in \(\mathbb{R}^3\). If \(M^2\subset \mathbb{R}^3\) is a closed embedded surface, the authors prove that there exist arbitrarily small \(C^\infty\) graphs \(M'\) such that\N\begin{enumerate}\N\item either the weak mean curvature flow of \(M'\) has only multiplicity-one spherical and cylindrical tangent flows until it goes extinct, \N\item or there exists a singular time \(T\) such that the tangent flow of \(M'\) at \(T\) either has multiplicity \(\geq 2\) or has a cylindrical end but is not a cylinder.\N\end{enumerate}\N\NThe main technical ingredient in proving the above main theorems is a long-time existence and uniqueness result for ancient one-sided flows which states that for any smooth self-shrinker \(\Sigma\subset\mathbb{R}^{n+1}\) that is either compact or asymptotically conical, there exists a unique ancient mean curvature flow \(t\mapsto \bar{M}(t)\) such that \(\bar{M}(t)\) is disjoint from \(\sqrt{-t}\Sigma\) and has entropy \(<2F(\Sigma)\) (the \(F\)-functional of \(\Sigma\)). This ancient solution \(\bar{M}(t)\) has several important properties: \N\begin{enumerate}\N\item it only has multiplicity-one generalized cylindrical singularities, \N\item it is smooth and star-shaped at \(t=0\), \N\item \(\frac{1}{\sqrt{t}}\bar{M}(t)\) converges smoothly on compact sets as \(t\to \infty\) to an outermost expander coming out of the cone at infinity of \(\Sigma\).\N\end{enumerate}\N\NTo prove the main theorem for \(M^3\subset\mathbb{R}^4\) with low entropy, the authors embed \(M\) in a local foliation \(\{M_s\}_{s\in (-1,1)}\) and flow the entire foliation simultaneously. The entropy bound implies that any tangent flow at the singularities must be compact or an asymptotically conical self-shrinker. Then the results on ancient one-sided flows are used to complete the proof and a density drop argument is required. For the case of embedded \(M^2\subset \mathbb{R}^3\), the argument is similar but a genus monotonicity argument is used to replace the density drop argument.